It is well-known that distinct exponential functions are linearly independent. What is known about the linear (in)dependence of the set of terms of an exponential polynomial? By exponential polynomial I mean a finite sum of terms of the form p_i(x) e^{a_i x} where each p_i is a polynomial. Here the exponential functions should be distinct, so the a_i are distinct; also the polynomials should be nonzero (that is, not identically zero). Under these conditions, is the set of functions p_1(x) e^{a_1 x}, ... , p_n(x) e^{a_n x} linearly independent?
2026-04-23 02:00:40.1776909640
Linear (in)dependence of distinct nonzero exponential polynomials
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The answer is yes. A proof for a more general resulting can be foundation in
http://scholar.google.at/scholar_url?url=https://www.degruyter.com/downloadpdf/j/dema.2008.41.issue-3/dema-2013-0091/dema-2013-0091.xml&hl=en&sa=X&scisig=AAGBfm0VysxlFjgoiWZgO1rxRweNaSSa4Q&nossl=1&oi=scholarr
Edit (2020-03-25): The link above ist broken. Take the following one. Demonstration mathematica